Find the side of the square.

Question

The problem I am proposing to solve was posed in a math contest for students of 17-18 years old, this month.

With the data in the picture, find the side of the square.

I did find the solution but it involves a long way across calculating the combined length of the red segments (not drawn in the original problem) and minimizing it for points on the segment of length $1$. With the diagonal we can get the side. I think there is a better and quicker way to solve it. I bring it here, maybe somebody could find this solution.

Answer 1

It is just a simple pythagoras theorem. You find the diagonal of the square $=\sqrt {50}$
Therefore the side of the square is $=5$

Answer 2

Assuming the red line means nothing, and $1$ is the length of the gray segment between the $2$ segment and the $5$ segment:

Turn the square (almost) $45^\circ$ around, so that the $5$ segment becomes horizontal, and place the corner of the square where the $5$ segment starts at the origin. The opposite corner of the square will be at $(7, 1)$.

What is the length of the diagonal of the square? Then what is its side length?

Answer 3

Rotate the square so that the $5$ and $2$ segments are horizontal:

Label the left most corner as $(0,0)$. Now hopefully you can see that the right most corner is at $(7,1)$.

From this, we can calculate the distance from the left most to the right most as

\begin{align}d^2&=7^2+1^2\\ &=49+1\\ &=50\\ d&=\sqrt{50}\end{align}

Now we can calculate the side length of the square as we know its diagonal length

\begin{align}s^2+s^2&=d^2\\ 2s^2&=\sqrt{50}^2\\ 2s^2&=50\\ s^2&=25\\ s&=\sqrt{25}\\ s&=5\end{align}